A Dutch and Spanish couple who have lived in Britain all their adult lives have told of their “devastation” after the Home Office refused their post-referendum application to have their two London-born children recognised as permanent residents of the country.

Jan-Dinant Schreuder and Monica Obiols, both 49, found themselves in a “bureaucratic nightmare” when they were told their 15-year-old son and 12-year-old daughter had to provide more evidence that they lived permanently with their parents.

“What evidence are they supposed to have? They don’t have council tax bills or proof of where they live. They are children,” said Obiols. “They have gone to school here all their lives, English is their mother tongue. I was just so shocked when we got the refusal letters.”

Their experience is the latest of many reported by the Guardian illustrating how red tape is heaping worry on families across the country.

In one recent case, a young man born to German parents in the UK was advised to take a citizenship test before he could get a British passport, because the tax authorities could not locate his mother’s tax records.

In another, a French scientist, who had worked in the UK for 20 years, and his British wife decided to quit the country after his application for permanent residency was rejected.

Obiols, a special needs teacher, has been living continuously in the UK since 1988, while Schreuder, a teacher in a secondary school, has been in the country since he was three.

Both their children have Spanish passports. Under EU law, nothing more needed to be done after they were born to establish their right to remain in the UK.

But the family became “panicked” after the referendum last June, thinking they needed to have paperwork in case they were ever asked to prove their status – for example, in a hospital or by an employer, said Schreuder.

In September, he decided to apply for indefinite leave to remain (ILR) as he had been in the country since 1970, three years before the UK joined the European Economic Community. Obiols applied for a permanent residency (PR) card for herself and their two children. She included both children’s birth certificates with her application.

The Home Office has now issued Schreuder with an ILR card and Obiols with a PR card. However, to the family’s horror, the Home Office sent separate notices to the two children notifying them that their applications for PR had been refused.

“You have not provided evidence that you have resided in the United Kingdom in accordance with those Regulations [European Economic Area immigration rules] for a continuous period of 5 years,” said the letter, signed on behalf of the secretary of state, Amber Rudd.

“It was surreal. I was so shocked that [it was] the children, who, probably of all people, were most deserving,” said Obiols. “It’s scary because people say, ‘oh, you can always leave the country if Brexit doesn’t work out’, but we can’t because the children are in school. We can’t disrupt their education and why should we. I feel quite devastated.”

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Schreuder said he did not understand the Home Office policy on children of parents deemed eligible to remain in the country.

“Their approach seems to be based on a weird assumption that parents would have children and then their children wouldn’t actually live with them – that they would send them off to another country to live with their grandparents or something. We are actually stunned,” he said.

Obiols said she included the children’s birth certificates with the application form and would now reapply, but felt she would need to “create evidence” that they lived at home.

Jan Doerfel, an immigration lawyer, said he believed the Home Office may have made a mistake under laws applying to children born after 1983. If Schreuder’s mother had obtained ILR documents for him when he was a boy, then his children are entitled to British citizenship.

“Even if he did not [have the documents], the mother would be considered settled once she had exercised treaty rights as a European citizen for five years,” Doerfel said.

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Having been in possession of the birth certificates of Obiols and Schreuder’s children, which showed they were born in the UK, and details about their parents’ status, the Home Office case worker should have notified the family that the children were likely to be British citizens, Doerfel added.

Schreuder came to Britain as a small boy when his doctor father and mother decided to move to the UK after a stint working and living in Kenya. “They met lots of British doctors there and liked them, and when they came back to Europe they decided they wanted to live in Britain, because they thought Holland was dull and boring by comparison. It was an adventure,” said Schreuder.

Theresa May, the prime minister, has been accused of using EU citizens as bargaining chips in Brexit negotiations. But after a strong campaign from grassroots organisations such as the3million, the European commission has said the issue will be one of the top priorities in the talks.

“If you had said to me before the referendum that I would be worrying about my future, worried about choices my children make for university, what fees they will pay etc, I would not have believed you. Now the things I considered completely safe are being decided by other people,” said Obiols.